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arX
iv:1
212.
1338
v1 [
cond
-mat
.sof
t] 6
Dec
201
2
Dynamics of cluster formation in driven dipolar
colloids dispersed on a monolayer
Sebastian Jager, Holger Stark, and Sabine H. L. Klapp
Institute of Theoretical Physics, Technical University Berlin,
Hardenbergstr. 36, 10623 Berlin, Germany
E-mail: [email protected]
Abstract. We report computer simulation results on the cluster formation of dipolar
colloidal particles driven by a rotating external field in a quasi-two-dimensional setup.
We focus on the interplay between permanent dipolar and hydrodynamic interactions
and its influence on the dynamic behavior of the particles. This includes their
individual as well as their collective motion. To investigate these characteristics,
we employ Brownian dynamics simulations of a finite system with and without
hydrodynamic interactions. Our results indicate that particularly the translation-
rotation coupling from the hydrodynamic interactions has a profound impact on the
clustering behavior.
Dynamics of cluster formation in driven dipolar colloids dispersed on a monolayer 2
1. Introduction
Systems of colloidal particles can organize into a wide variety of different structures,
both under equilibrium conditions (“self-assembly”) and in nonequilibrium. Prime
examples of nonequilibrium structure formation are lane formation of charged colloids
[1, 2], shear banding of rod-like particles [3], the coiling up of chains of magnetic colloids
[4], metachronal waves in driven colloids [5], and the formation of colloidal caterpillars
[6].
Indeed, magnetic particles form a particular subset of colloidal systems, since
they can be easily manipulated by external magnetic fields. The interplay of particle-
field interactions and the anisotropic (dipole-dipole) magnetic interactions between the
particles then leads to a wide variety of structures such as chains [7, 8, 9], layers
[9, 8, 10, 11, 12], and intricate honeycomb-like structures [13, 14, 15].
In this study we want to focus on the pattern formation of dipolar particles that are
driven by a rotating field. In three-dimensional systems dipolar particles exposed to such
a field tend to form layers (biaxial field) [16, 11] or membrane-like structures (triaxial
field) [13, 14]. In a two-dimensional geometry, in which the external field rotates in the
plane of the particles, the dipoles tend to agglomerate into clusters. This phenomenon
has been observed experimentally [17, 18] as well as in computer simulations that were
performed by two of the authors of this study [19]. Clustering does not only occur for
particles carrying a permanent dipole moment, but also for particles with an induced
dipole moment [20, 21, 22].
At suitable fields strengths and frequencies, the (permanently) dipolar particles
perform a synchronous rotation with the field. This motion gives rise to an attractive,
long-range particle interaction (∝ −1/r3), which induces a first-order phase transition
between a dilute and a denser phase (in an infinitely extended system). The observed
cluster formation then corresponds to spinodal decomposition inside of the coexistence
region [19].
It has been shown that hydrodynamic interactions play a crucial role in numerous
colloidal systems [23, 24, 25, 26]. The influence of hydrodynamic interactions on the
clustering phenomenon was, however, only shortly touched upon in [19]. Here, in this
study, we want to look into the effects that an implicit solvent has on the cluster
formation and the dynamic behavior of the particles in more detail.
The main tool we use are Brownian dynamics computer simulations with and
without hydrodynamic interactions. The particles are modeled by a quasi-two-
dimensional system of soft spheres with permanent dipole moments. This means that
the dipoles can rotate freely in all the spatial directions, while the translational motion
is restricted to a two-dimensional plane. In our hydrodynamic simulations, we not only
account for the translational hydrodynamic couplings, but for all the couplings between
the translational and rotational motions of the particles. To understand the influence
of these different couplings is one of the goals of this study. Contrary to [19], we here
consider a finite system size, i.e., we do not employ periodic boundary conditions.
Dynamics of cluster formation in driven dipolar colloids dispersed on a monolayer 3
This paper is organized as follows: After introducing the model and the simulation
techniques, we discuss the influence of hydrodynamic interactions on the regions of
cluster formation in the field strength-frequency domain. To understand the differences
stemming from the solvent interactions, we also consider the magnetization and
synchronization behavior of the particles. In a next step, we investigate the dynamics
of the clusters and their formation. Finally, we examine the structure of the clusters
with respect to their internal order. The paper is then closed with a brief summary and
conclusions.
2. Model and simulation methods
In this study, we consider a quasi-two-dimensional system of dipolar colloidal particles
that are immersed in a solvent. As a model for the colloids we use a dipolar soft sphere
(DSS) potential, which is comprised of a repulsive part U rep and a point dipole-dipole
interaction part UD:
UDSS(rij,µi,µj) = U rep(rij) + UD(rij ,µi,µj) (1)
In (1), rij is the vector between the positions of the particles i and j, rij its absolute
value, and µi is the dipole moment of the ith particle. The dipolar and repulsive
interaction potentials are given by
UD(rij,µi,µj) = −3(rij · µi)(rij · µj)
r5ij+
µi · µj
r3ij(2)
and [27]
U rep(r) = USS(r)− USS(rc) + (rc − r)dUSS
dr(rc), (3)
respectively. Here, U rep is the shifted soft sphere potential, where
USS(r) = 4ǫ
(
σ
rij
)12
(4)
is the unshifted soft sphere (SS) potential for particles of diameter σ.
We investigate the system by making use of Brownian dynamics (BD) simulations
with and without hydrodynamic interactions (HIs). Specifically, the positions of the
particles are evolved in time via [28, 29, 30]
ri(t+∆t) = ri(t) +1
kBT
∑
j
DTTij Fj∆t
+1
kBT
∑
j
DTRij Tj∆t +Ri(D,∆t) (5)
Dynamics of cluster formation in driven dipolar colloids dispersed on a monolayer 4
whereas their orientations ei = µi/µ evolve according to
ei(t +∆t) = ei(t) +
(
1
kBT
∑
j
DRTij Fj∆t
+1
kBT
∑
j
DRijTj∆t
)
× ei(t) +Ri(D,∆t)× ei(t). (6)
The forces and torques in (5) and (6) are given by
Fi = −∇ri
∑
j 6=j
UDSS(rij,µi,µj), (7)
Ti = TDSSi +Text
i , (8)
where
TDSSi = −µi ×∇µi
∑
j 6=j
UDSS(rij ,µi,µj), (9)
Texti = µi ×Bext. (10)
In (10), Bext denotes an external field that is homogeneous and rotates in the plane of
the dipolar monolayer. Specifically,
Bext(t) = B0(ex cosω0t+ ey sinω0t), (11)
where ω0 is the frequency of the field and B0 its strength.
In (5) and (6), DTTij , DTR
ij , DRTij , and DRR
ij are subtensors of DTT, DTR, DRT, and
DRR. The latter are subtensors of the grand diffusion tensor
D =
(
DTT DTR
DRT DRR
)
(12)
and will be specified below. The random displacements in (5) and (6) behave according
to
〈Ri〉 = 0, (13)
〈Ri(∆t)Rj(∆t)〉 = 2Dij∆t. (14)
The actual calculation of these displacements can be done by evaluating
R =√2∆tL · ξ, (15)
where R is the vector comprised of all the Ri, ξ is a vector of normally distributed
random numbers, and L is a lower triangular matrix which satisfies
D = L · LT . (16)
Dynamics of cluster formation in driven dipolar colloids dispersed on a monolayer 5
In the present study, we take the HIs into account up to third order in the inverse
particle distance, which corresponds to a far field approximation. The tensors DTT,
DTR, DRT, and DRR are then given by [29, 30, 26]
DTTii =
kBT
6πησI, (17)
DTTij =
kBT
8πη
1
rij
[
(I+ rij rij) +2σ2
3r2ij(I− 3rij rij)
]
, (18)
DRTii = DTR
ii = 0, (19)
DTRij = D
RT†ji =
kBT
8πη
1
r2ijǫrij, (20)
DRRii =
kBT
8πησ3I, (21)
DRRij =
kBT
16πη
1
r3ij(3rij rij − I), (22)
where η is the viscosity of the solvent and ǫ the Levi-Civita density, which satifies
ǫ123 = ǫ231 = ǫ312 = 1, ǫ321 = ǫ213 = ǫ132 = −1, and is equal to zero for other choices
of the indices. These tensors describe the different hydrodynamic couplings, i.e., the
coupling between the translational motion of the particles (TT), between the translation
and the rotational motion (TR) and vice versa (RT), and the coupling between the
rotational motion (RR). The specific tensor given in (17) and (18), i.e., DTT, is the well
known Rotne-Prager tensor [31].
Note that by setting to zero all the hydrodynamic coupling tensors involving
different particles i 6= j in (17)-(22), one arrives at the standard algorithm for BD
simulations without HIs [32].
Considering the equation of motion (5), we can see that the TR coupling relates all
the torques acting on the particles, i.e., their rotational motions, to all the translational
motions of the particles. Physically, the TR coupling describes how the translational
motion of the particles is affected by the flow fields in the solvent that are caused by
the rotations of the particles. Indeed, as we will see later, the TR/RT coupling is
particularly relevant for the overall dynamical behavior of the system. It is therefore
instructive to briefly illustrate the implications of this coupling for a simple two-particle
system (for details, see [30, 26]).
To this end, let us consider two particles that are located at a distance from each
other on the x-axis in a right-handed coordinate system. Due to the TR coupling,
anticlockwise rotation (following the application of a torque) of the particle at the
larger value of x results in the other particle moving in the negative y-direction. By
realizing that the flow fields follow the rotation of the particle, this process can be easily
understood. On the other hand, an anticlockwise rotation of the particle at smaller
values of x causes the other particle to move into the positive y-direction (for a sketch,
see [30]).
In this study, we consider N = 324 particles in a simulation box that is bounded
by soft walls (cf. [33]), i.e., we do not use periodic boundary conditions. Therefore, it
Dynamics of cluster formation in driven dipolar colloids dispersed on a monolayer 6
is not necessary to use special techniques (e.g., Ewald sums) to treat the long-range
dipole-dipole interactions. The forces and torques can be calculated directly via (2).
For convenience, we make use of the following reduced units: Field strength
B∗0 = (σ3/ǫ)1/2B0; frequencies of the field ω∗
0 = ω0σ2/DT
0 , where DT0 = kBT/3πησ is
the translational diffusion constant; dipole moment µ∗ = (ǫσ3)−1/2µ; time t∗ = tDT0 /σ
2;
temperature T ∗ = kBT/ǫ; position r∗ = r/σ. In the following we specialize to systems
at temperature T ∗ = 1 and of dipole moment µ∗ = 3. This choice corresponds to a
dipolar coupling strength of λ = µ∗2/T ∗ = 9 that is sufficiently large to enable the
system to form clusters for suitable field strengths and frequencies [19]. The density of
the particles in the simulation box is of no importance in the investigated systems, since
the particles typically agglomerate into a single cluster.
3. Dynamics on the particle level
Applying a rotating field in the plane of a dipolar monolayer can cause the dipolar
particles to agglomerate into two-dimensional clusters [19]. This agglomeration is caused
by a synchronization phenomenon: At suitable field strengths and frequencies, the
particles follow the field at the same phase difference resulting in an effective interparticle
interaction of the form [34, 19]
U ID(rij) = − µ2
2r3ij. (23)
Equation (23) can be arrived at by averaging the dipole-dipole potential over one
rotational period of the field under the assumption that µi(t) = µj(t) = µ0[ex cos(ω0t+
δ) + ey sin(ω0t + δ)]. Here, δ is some phase difference. A further crucial assumption
in the derivation of (23) is that the translational motion of the particles during one
rotational period of the field is negligible.
The potential (23) is attractive, which leads to the aforementioned cluster
formation. Moreover, in an infinitely extended system, i.e., in the thermodynamic limit,
the potential (23) gives rise to a first order phase transition [19].
Figure 1. Snapshots of systems at B∗
0= 50 and ω∗
0= 240 after clusters have formed.
(a) Without and (b) with HIs.
Dynamics of cluster formation in driven dipolar colloids dispersed on a monolayer 7
Simulation snapshots of systems after clusters have formed can be seen in figures
1(a) and (b). The former shows the snapshot of a system (B∗0 = 50, ω∗
0 = 240), whose
particles do not interact via HIs, while HIs are included in the system associated with
the latter snapshot. Since the formation of clusters can be observed in both the systems,
we note as a first result that cluster formation is not prevented by the presence of HIs.
As already noted in [19], this is not a priori clear, since HIs can induce additional
motion in a nonequilibrium system and the averaged potential (23) is only valid as an
approximation to the true interparticle interaction if the translational motion of the
particles during one rotational period of the field is small. This is certainly the case for
the system without HIs: Inspecting the mean squared displacements (averaged over all
the particles) at B∗0 = 50 and ω∗
0 = 240, we find that the field rotates about 30 times
before a particle transverses a distance of its own diameter σ.
The driving frequency ω∗0 = 240 chosen in figure 1 corresponds to ω0 ≈ 54 MHz,
if we assume the values of the diffusion constant (DT0 ≈ 38 µm2/s) and particle size
(σ = 13 nm) that are given in [29] for a ferrofluid.
Note that the cluster formation in infinitely extended quasi-two-dimensional
systems corresponds to spinodal decomposition within the coexistence region of a phase
transition [19]. This is not exactly true for the cluster formation we observe in the
present study. The finite systems considered here do not undergo a phase transition.
However, the used system is well suited as a model system to investigate the influence
of the HIs on the dynamic behavior of the individual clusters.
In the following we ask how the collective rotational behavior of the particles
changes if solvent-mediated interactions are taken into account. First, consider figure
2, which shows whether cluster formation occurs for selected state points in the field
strength-frequency domain. Presented are results for both the cases with and without
HIs included. Compared to the simple BD system cluster formation breaks down at
smaller frequencies ω∗0 in the hydrodynamically interacting system. In the former, cluster
formation can be observed up to ω∗0 ≈ 450 (at B∗
0 = 50) while cluster formation ceases
at ω∗0 ≈ 350 when HIs are present.
To understand the breakdown of cluster formation in more detail, we now examine
the rotational motion of the particles. As explained above, synchronized rotation is
necessary for (23) to hold, i.e., is a prerequisite for cluster formation. Figure 3 shows
the absolute value of the magnetization normalized with respect to its saturation value
M(t)
M0
=1
Nµ
⟨∣
∣
∣
∣
∣
N∑
i=1
µi
∣
∣
∣
∣
∣
⟩
(24)
over the driving frequency ω∗0 of a system [B∗
0 = 50, cf. Fig. 2] with all the hydrodynamic
couplings and without HIs included. The magnetization indicates how aligned the
particles are in a given state, i.e., indicates if they follow the field. As can be seen,
the magnetization starts with values close to one for both the systems, corresponding
to an aligned state. At ω∗0 ≈ 270 the magnetization begins to drop for the system that
Dynamics of cluster formation in driven dipolar colloids dispersed on a monolayer 8
20
40
60
80
B∗ 0
With HI
0 100 200 300 400 500 600ω ∗0
20
40
60
80
B∗ 0
No HI
Clustered
Not clustered
Figure 2. Cluster formation (top) with and (bottom) without HIs in the field strength-
frequency domain. Squares/circles indicate where cluster formation occurs/does not
occur. The systems are at temperature T ∗ = 1 and dipole moment µ∗ = 3.
includes HIs. The magnetization in the system without HIs remains at M/M0 ≈ 1 up
to ω∗0 ≈ 420.
0 100 200 300 400 500 600ω ∗0
0.0
0.2
0.4
0.6
0.8
1.0
M/M
0
All HI
No HI
Figure 3. Magnetization normalized with respect to its saturation value over the
driving frequency of the external field for a system with and without HIs. The fields
used are of strength B∗
0= 50. The vertical line indicate where cluster formation ceases
with (dashed) and without (solid) HIs.
This magnetization behavior implies that the particles in the hydrodynamically
interacting system are less aligned with each other for ω∗0 & 270. In particular, the
synchronization breaks down at lower frequencies resulting in a premature breakdown
of cluster formation.
Dynamics of cluster formation in driven dipolar colloids dispersed on a monolayer 9
4. Cluster dynamics
We now turn to the dynamics of the entire cluster. In [19], it was already shown that in
systems with HIs, clusters form at a significantly faster pace (as compared to systems
without HIs). In this section, we aim to investigate the changes induced by HIs in more
detail.
0 2 4 6 8 10 12 14 16 18t ∗
9
10
11
12
13
14
15
d [σ]
All HI
No RT/TR
No RR
No HI
Figure 4. Mean distance between the particles over time d in a system (B∗
0= 50, ω∗
0=
240) with all hydrodynamic interactions included (All HI), without the hydrodynamic
RT/TR coupling (No RT/TR), without the hydrodynamic RR coupling (No RR), and
without all HIs (No HI).
First, consider figure 4, which shows the mean distance
d(t) =2
N(N − 1)
N∑
i=1
∑
j<i
rij(t) (25)
between the particles for a system at B∗0 = 50 and ω∗
0 = 240. These particular field
parameters were chosen for three reasons: First, the frequency is sufficiently high
to ensure that the effective potential (23) describes the interparticle interaction well.
Second, as seen in figure 2, cluster formation occurs for both a hydrodynamically as
well as a not hydrodynamically interacting system. Third, the magnetization of the
systems with these choices of ω∗0 and B∗
0 is maximal and identical irrespective of the
presence of HIs [cf. figure 3].
Specifically, figure 4 shows the evolution of d over time for particles interacting via
HIs including all the hydrodynamic couplings, for particles lacking the hydrodynamic
RT/TR coupling, particles lacking the RR coupling, and particles not interacting via
HIs at all. In all the cases, d assumes a constant minimal value at long times. To
understand this, recall that we consider a single simulation box filled with particles here.
In an infinite system, the cluster would keep growing in time indefinitely with a power
law behavior [35, 36, 37], since the process corresponds to spinodal decomposition [19].
Here, however, the growth process stops once all the particles have been incorporated
into the cluster and a stationary state is reached.
Dynamics of cluster formation in driven dipolar colloids dispersed on a monolayer 10
In the systems that include solvent-mediated interactions (All HI, No RT/TR,
No RR), the value of d drops significantly faster than in the case without any HIs.
Consequently, the average distance between the particles decreases faster, which means
that the cluster formation process is sped up. The acceleration is neither influenced by
the lack of the RT/TR nor the RR coupling, which implies that the TT coupling alone
is responsible for this effect. Note, however, that the lack of the presence of the RT/TR
coupling expresses itself by a different value of d at long times [see figure 4].
1 2 3 4 5 6 7 8 9 10r ∗c
0
5
10
15
20
ω∗ cl
All HI
No RT/TR
Figure 5. Mean angular frequency ω∗
claround the cluster center of the particles over
distance from particle center r∗cfor a system at B∗
0= 50 and ω∗
0= 240. Shown are
values for a system interacting via all the hydrodynamic couplings and for a system
lacking the RT/TR coupling.
The RT/TR coupling does have another interesting influence on the dynamic
behavior of the particles. In figure 5, the mean angular frequency of the particles with
respect to the cluster center over the distance from the center is shown for a system
at B∗0 = 50 and ω∗
0 = 240. Values for the hydrodynamically interacting case with all
the couplings included as well as the case lacking the RT/TR coupling are presented.
In the system that includes the RT/TR coupling, the angular velocity of the particles
differs from zero at all the displayed distances from the cluster center r∗c . Hence, the
particles perform a rotation around the cluster center in the rotational direction of the
external field. The system lacking the RT/TR coupling does not show such a rotational
behavior. As can be seen in figure 5, the mean angular frequency of the particle around
the cluster center is essentially zero at all distances.
This collective rotation is caused by the individual, field-driven rotations of the
particles. The rotational motion of the particles creates a flow field that induces
translational motion in all the other particles [see the argument given below (17)-(22)].
Therefore, the TR coupling alone is responsible for this behavior. The RT coupling does
not contribute in any way to the cluster rotation.
As we have seen in the previous section, HIs result in a premature breakdown of
cluster formation, i.e., a breakdown at smaller driving frequencies of the field (relative
to the case without HIs). It stands to reason that the cluster rotation induced by
the HIs has a significant influence on this behavior. The particles perform additional
translational motion (around the cluster), which makes the effective potential (23) less
accurate as a description for the interparticle interaction at a given driving frequency.
The more the particles move during one period of the field, the less does the effective
Dynamics of cluster formation in driven dipolar colloids dispersed on a monolayer 11
potential capture the actual interaction between the particles.
Finally, note that the RR coupling does not seem to have any significant influence
on the dynamic behavior of the cluster. That is, it does not contribute to the accelerated
cluster formation or the cluster rotation. The former is illustrated by figure 4, which
shows that the mean distance d behaves essentially identically to the system with all the
HIs included. The fact that the cluster rotation is not influenced by the RR coupling
can be seen in figure 5. Despite the presence of the RR coupling, the cluster does not
rotate if the RT/TR coupling is absent.
5. Internal structure of the cluster
In a recent experimental study, Weddemann et al. [17] showed the existence of cluster
formation in two-dimensional systems of (permanently) dipolar particles that are driven
by a rotating external field. In particular, the authors of [17] observed the formation of
hexagonally ordered particle agglomerates in their experiments.
The general clustering phenomenon of dipolar particles exposed to rotating fields
was later identified as spinodal decomposition in a simulation study by two of the
authors of the present study [19]. However, to reproduce clusters of hexagonal order,
very low temperatures inside of the two-phase coexistence region had to be considered
in computer simulations. It was conjectured that the hexagonally ordered clusters occur
in the vapor-solid coexistence region, i.e., at coupling strengths above the ones related
to the vapor-liquid region.
As shown in the previous section, HIs can have a significant influence on the
collective motion of the particles. Here, in this section, we want to investigate, whether
HIs preserve the internal cluster structure. Despite experimental evidence of the
hexagonal order, this fact is debatable since it remains unclear to what degree the
rotational motion of the magnetic particles in the experimental work [17] follows the
dipole moment.
In order to gain insight into the emergent (hexagonal) structures in the present,
finite systems, we consider the bond order parameter
ψ6 =1
N
N∑
n=1
1
|Nn|
∣
∣
∣
∣
∣
∑
k∈Nn
exp(i6πφnk)
∣
∣
∣
∣
∣
(26)
at different dipolar coupling strengths λ. Here, Nn is the set of nearest neighbors of
particle n, which consists of particles that are closer to particle n than the distance of
the first minimum in the pair correlation function of the system. The systems considered
in the following are of sufficiently high coupling strength ensuring that cluster formation
does indeed occur.
In figure 6, ψ6 as function of the dipolar coupling strength λ for systems (B∗0 = 50)
with and without HIs at ω∗0 = 240 and 300 is shown. We note that ψ6 is essentially
independent of time in the stationary situation, where all the particles are part of
the cluster. At ω∗0 = 300, the bond order parameter increases with λ for both the
Dynamics of cluster formation in driven dipolar colloids dispersed on a monolayer 12
0.00.20.40.60.81.0
ψ6
ω ∗0 =240
8 10 12 14 16λ
0.00.20.40.60.81.0
ψ6
ω ∗0 =300
No HI
All HI
Figure 6. Bond order parameter ψ6 over coupling strength λ for a system (B∗
0= 50,
T ∗ = 1) with and without HIs at (top) ω∗
0= 240 and (bottom) ω∗
0= 300.
hydrodynamically as well as the not hydrodynamically interacting system. If HIs are
not present, an increase from ψ6 ≈ 0.52 at λ = 7.02 to ψ6 ≈ 0.72 at λ = 16 can
be observed. This increase qualitatively agrees with the one observed in the Langevin
dynamics simulations in [19]. In the system with HIs included, the bond order parameter
ψ6 increases considerably less with increasing coupling strength. However, there is still
significant order in the system.
In the system with ω∗0 = 240 and HIs included, ψ6 behaves similarly to the case
with ω∗0 = 300 and HIs. If these interactions are not taken into account, however, less
order than in the ω∗0 = 300 system can be observed. The smaller frequency allows for
more translational motion during one rotational period of the field, resulting in more
spatial inhomogeneity.
Now consider figure 7, which shows the pair correlation functions of a system
(B∗0 = 50, ω∗
0 = 300) with and without HIs included. The pair correlation function of
the not hydrodynamically interacting system shows a double peaked maximum after the
first minimum, which is typical for a hexagonally ordered system. The hydrodynamically
interacting system, on the other hand, does not have this feature. Hence, the particles
tend to have six angularly equally distributed neighbors as shown by the value of ψ6
[cf. figure 6], but seem to lack the long-range positional order of a hexagonally structured
system. Further, the extrema are much more pronounced in the system without HIs,
indicating a more ordered state.
In conclusion, in both systems in figure 6, the HIs tend to weaken the hexagonal (or
hexatic) order present in the system. The lower value of ψ6 in systems with HIs can be
explained by the collective cluster rotation induced by the hydrodynamic TR coupling.
The particles rotate around the cluster center and do not stay at fixed lattice sites. This
behavior results in a reduction of the bond order parameter and the hexagonal structure
in the system.
Dynamics of cluster formation in driven dipolar colloids dispersed on a monolayer 13
0 1 2 3 4r ∗
0
2
4
6
8
g(r∗ )
All HI
No HI
Figure 7. Pair correlation functions of systems at B∗
0= 50 and ω∗
0= 300 with and
without HIs included.
6. Conclusions
In this study, we have investigated the influence of the solvent on the dynamic behavior
of rotationally driven dipolar particles.
The cluster dynamics and formation is influenced in three major ways by HIs. First,
HIs accelerate the cluster formation process. While the particle approach each other,
flow fields are created that drag other particles along. Second, we have shown HIs to
induce a collective cluster rotation. Without HIs no such rotation can be observed. The
driven rotating particles pull the solvent with them in their rotational motion, which
results in the translational motion of the particles around the cluster center. As a last
major point, we have shown cluster formation to cease at lower driving frequencies of
the field. We attribute this to an earlier breakdown of synchronization if HIs are present
and an increase in translational motion due to the collective cluster rotation.
Moreover, we have studied the influence of HIs on the internal structure of the
clusters. Our results indicate that HIs tend to decrease the hexagonal order in the
systems.
Recently, one of the authors of this study has conducted an investigation of the
structure of a closely related system [38]. The system consisted of particles that only
interact via HIs, are confined to a monolayer, and have fixed angular velocities. It was
found that hexagonal particle agglomerations rotate, with the hexagonal order melting
and recrystallizing periodically. In the system focused on in the present study, no such
phenomenon could be observed. We attribute this to the presence of dipolar interactions
in our system and a lack of Brownian motion in [38].
Another study has recently investigated the interplay of confinements and HIs [39].
It was shown that particle rotations can be utilized to create directed translational
motion in colloids in specific geometries. This result suggests that it might be interesting
to examine the effects of different confinements on the dipolar systems considered here.
In conclusion, we have shown that HIs have a considerable influence on the
Dynamics of cluster formation in driven dipolar colloids dispersed on a monolayer 14
formation and the dynamics of clusters of driven dipolar colloidal particles in a two-
dimensional geometry. Usually, HIs seem to affect colloidal nonequilibrium clustering
phenomena less significantly than in the present system. As an example, consider the
process of colloidal gelation in two-dimensional Lennard-Jones systems [40]. In contrast
to our system, the agglomeration of the particles in [40] is only marginally influenced by
the HIs. In general, however, HIs can significantly alter nonequilibrium processes. For
instance, HIs can enhance ratchet effects [41, 42] or synchronize the motion of eukaryotic
[43] or bacterial [44] flagella.
Acknowledgments
We gratefully acknowledge financial support from the DFG within the research
training group RTG 1558 Nonequilibrium Collective Dynamics in Condensed Matter
and Biological Systems, project B1.
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